To solve the problem of finding the mean free path of \( O_2 \) at high altitude with a pressure of 100 mm and a temperature of 200°C, we can follow these steps:
### Step 1: Understand the Mean Free Path Formula
The mean free path (\( \lambda \)) can be calculated using the formula:
\[
\lambda = \frac{kT}{\sqrt{2} \pi d^2 p}
\]
Where:
- \( k \) is the Boltzmann constant,
- \( T \) is the absolute temperature in Kelvin,
- \( d \) is the diameter of the gas molecule,
- \( p \) is the pressure.
### Step 2: Identify the Relationship Between Conditions
From the formula, we can see that the mean free path is directly proportional to temperature and inversely proportional to pressure. Therefore, we can use the relationship:
\[
\frac{\lambda_2}{\lambda_1} = \frac{T_2}{T_1} \cdot \frac{P_1}{P_2}
\]
Where:
- \( \lambda_1 \) is the mean free path at the initial conditions,
- \( \lambda_2 \) is the mean free path at the new conditions,
- \( T_1 \) and \( T_2 \) are the initial and final temperatures in Kelvin,
- \( P_1 \) and \( P_2 \) are the initial and final pressures.
### Step 3: Convert Temperatures to Kelvin
Convert the given temperatures from Celsius to Kelvin:
- \( T_1 = 27°C = 27 + 273 = 300 \, K \)
- \( T_2 = 200°C = 200 + 273 = 473 \, K \)
### Step 4: Identify Pressures
Convert the pressures:
- \( P_1 = 1 \, atm = 760 \, mmHg \)
- \( P_2 = 100 \, mmHg \)
### Step 5: Substitute Values into the Equation
Now we can substitute the known values into the equation:
\[
\frac{\lambda_2}{10^{-5}} = \frac{473}{300} \cdot \frac{760}{100}
\]
### Step 6: Calculate the Right Side
Calculate the right-hand side:
1. Calculate \( \frac{473}{300} \):
\[
\frac{473}{300} \approx 1.5767
\]
2. Calculate \( \frac{760}{100} \):
\[
\frac{760}{100} = 7.6
\]
3. Multiply these two results:
\[
1.5767 \cdot 7.6 \approx 11.9827
\]
### Step 7: Solve for \( \lambda_2 \)
Now, substituting back to find \( \lambda_2 \):
\[
\lambda_2 = 11.9827 \cdot 10^{-5} \approx 1.19827 \cdot 10^{-4} \, cm \approx 1.2 \cdot 10^{-4} \, cm
\]
### Final Answer
Thus, the mean free path of \( O_2 \) at high altitude with a pressure of 100 mm and a temperature of 200°C is approximately:
\[
\lambda_2 \approx 1.2 \times 10^{-4} \, cm
\]
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